Ornstein-Uhlenbeck Neural Input

A single neuron in your brain is never quiet. Thousands of other neurons are constantly tapping it with tiny inputs, like standing in a packed crowd where everyone keeps gently bumping you. No single bump matters, but together they shove you around at random. Here is the twist: each bump does not vanish instantly — it lingers for a few milliseconds before fading. So the push you feel right now is a little bit like the push a moment ago. That "memory" in the randomness is what scientists call coloured noise, and it is different from pure static (white noise), where every instant is completely unrelated to the last.

The wandering input current $I_s$ has three personalities. It has an average value $\mu$ — the steady drive it tends to sit near. It jitters around that average by a typical amount $\sigma$ (the spread, or standard deviation). And the new ingredient is the correlation time $\tau_s$ — how long one bump lingers before it is forgotten. In words: the input is always gently pulled back toward its mean $\mu$, at a speed set by $\tau_s$, while fresh random kicks keep knocking it away. Worked number: for a fast AMPA synapse $\tau_s\approx 3$ ms, so a fluctuation now is mostly forgotten about 3 ms later — short, but not instant.

Putting "pull-back plus random kick" into one line gives the Ornstein-Uhlenbeck equation $\tau_s\frac{dI_s}{dt}=-(I_s-\mu)+\sigma\sqrt{2\tau_s}\,\xi(t)$, where $\xi(t)$ is Gaussian white noise. The deterministic term $-(I_s-\mu)$ is the spring pulling the current home; the $\sigma\sqrt{2\tau_s}\,\xi$ term is the random drive. This single rule produces an exponential autocorrelation $C(\tau)=\sigma^2 e^{-|\tau|/\tau_s}$ (memory that decays smoothly) and a Lorentzian power spectrum $S(f)=\frac{2\sigma^2\tau_s}{1+(2\pi f\tau_s)^2}$ — lots of power at low frequencies, rolling off above $f\sim 1/(2\pi\tau_s)$. In the limit $\tau_s\to 0$ the memory disappears and you recover ordinary white noise. Slider map: Input I sets the mean $\mu$, Parameter 1 sets the noise size $\sigma$, Parameter 2 sets the correlation time $\tau_s$, and T sim sets how long you watch.

(1) Push Parameter 2 ($\tau_s$) toward zero and watch the trace turn into jagged, hairy white-noise hash — no memory. (2) Raise $\tau_s$ and watch the same noise become smooth, slow, drifting waves. (3) Crank up Parameter 1 ($\sigma$) and see the spread widen while the average stays parked on $\mu$ — proof that $\mu$ and $\sigma$ are independent knobs.

Sum of many Poisson presynaptic spike trains convolved with alpha-function synaptic kernels → Gaussian process (CLT). The OU process is the simplest such Gaussian process with exponential autocorrelation (= alpha-function kernel). τ_s corresponds to synaptic time constant.

Diffusion (OU) valid when many spikes per τ_s: ν_pre × N_syn × τ_s >> 1. For AMPA (τ_s=3ms): need ν_pre × N_syn >> 333 spikes/s. With N_syn=100: ν_pre >> 3.3 Hz — easily met. At low rates/few synapses: use shot noise model instead.

LIF + white noise (τ_s→0): Siegert formula applies exactly. LIF + coloured noise (finite τ_s): firing rate R ≈ R_white(1 - (τ_s/τ_m) × correction term). Correction increases R (coloured noise slightly more efficient at driving spikes than white noise of same variance).